Problem: Divide the following complex numbers: $\dfrac{12 e^{5\pi i / 4}}{4 e^{19\pi i / 12}}$ (The dividend is plotted in blue and the divisor in plotted in green. Your current answer will be plotted orange.)
Explanation: Dividing complex numbers in polar forms can be done by dividing the radii and subtracting the angles. The first number ( $12 e^{5\pi i / 4}$ ) has angle $\frac{5}{4}\pi$ and radius 12. The second number ( $4 e^{19\pi i / 12}$ ) has angle $\frac{19}{12}\pi$ and radius 4. The radius of the result will be $\frac{12}{4}$ , which is 3. The difference of the angles is $\frac{5}{4}\pi - \frac{19}{12}\pi = -\frac{1}{3}\pi$ The angle $-\frac{1}{3}\pi$ is negative. A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $-\frac{1}{3}\pi + 2 \pi = \frac{5}{3}\pi$ The radius of the result is $3$ and the angle of the result is $\frac{5}{3}\pi$.